## The Rule of 72 Today I found out about the Rule of 72, which is a very easy way to calculate in your head how long it will take to double your money or debt based on a given fixed interest rate, assuming the interest is annually compounded.

Use of the Rule of 72 is very simple.  All you have to do is divide 72 by the interest rate.  The resulting number is the number of years it will take for the amount to double, given that fixed interest rate.  For example:  if you invest \$10,000 in a CD paying 4% compounded annually, it would take about 72/4 = 18 years to turn that into \$20,000.  On the flip side, if you have some amount of debt, say \$30,000 in student loans, at a 5% interest rate which you don’t make payments on, it will take 72/5 =  14.4 years for the amount owed to double to \$60,000.

You can also run the calculation the other way, if you want to determine what interest rate you’d need to double your money in a given amount of time.  For instance: if you have \$20,000 in savings and would like to double it in the next 10 years without adding anything to it, you’d need an interest rate of around 72/10 = 7.2%.

You can, of course, also use the Rule of 72 to calculate the effect of inflation on your money that you don’t invest.  So, if the annual inflation rate is at 2%, for instance, then in 72/2 = 36 years, your money that you didn’t invest will be worth half what it is today.

As you can see from the following table, the Rule of 72 is remarkably accurate:

 Return % Rule of 72 Years Actual Years 3% 24 23.45 4% 18 17.673 5% 14.4 14.21 6% 12 11.896 7% 10.3 10.24 8% 9 9.006 9% 8 8.04 10% 7.2 7.273

For those curious, how the Rule of 72 works is as follows (warning: there be math ahead; skip to the Bonus Factoids if you got a headache just from reading the word “math”) 😉 :  we start with the general formula for annually compounded interest: P(1+r)Y where Y is the number of years, P is the principle and r is the interest rate.  Now we want to see when it will double, so we modify it such that: 2P = P(1+r)Y

Now the exact principle doesn’t really matter here, we just want to know when it will double, so next we simplify the problem and solve for Y, so that: Y = ln(2) / ln(1+r)

Now we simplify that to Y = K / r, where  (K /r) = (ln(2)/ln(1+r)) and K will be some number that will result in a fairly accurate outcome given a certain range of values of r.

To begin with, we’ll see what value of K would work for a 10% interest rate:

Step 1: ln(2) / ln(1 + r) = K / r

Step 2: ln(2) / ln(1 + .1) = K / 0.1

Step 3: K = [ln(2) / ln(1.1)] * 0.1

Solution: K = .727

So here we see that the number we get out to be divided by the interest rate in the Rule of 72 is, not surprisingly, really close to 72, namely: 72.7.  Doing a similar calculation of 5% then results in .7103, so 71.03 when used to divide by the interest rate.

If you were to do the math for a wide range of commonly used interest rates, you’ll see that K always hovers reasonably close to 72, which was possibly picked over 71 or 73 or the like due to the fact that 72 has many small divisors that are in the range of commonly used interest rates: 1, 2, 3, 4, 6, 8, 9, and 12, and within whose range the Rule of 72 is quite accurate.  The Rule of 72 though does start to break down as you get to extremely high rates, such as 100%, where the Rule of 72 gives you .72 years, which is 28% off of the actual value of doubling in one year exactly.

Bonus Facts:

• There is also a “Rule of 69” that is derived and used in a similar fashion to Rule of 72, except that it is used to calculate doubling when the interest is compounded continually, rather than annually.  In this case, 69 is chosen because, when you work the math, compounding daily for typical interest rates comes out to around 69-70 and compounding daily is a reasonable approximation for compounding continually.
• The earliest reference to the Rule of 72 is from Summa de Arithmetica which was written around 1494 in Venice by Luca Pacioli.  In this work, he uses the rule without deriving it, so it is assumed that the rule was already well known at that time: (rough translation of that part of the work): “In wanting to know for any percentage, in how many years the capital will be doubled, you bring to mind the Rule of 72, which you always divide by the interest, and the result is in how many years it will be doubled. Example: When the interest is 6 percent per year, I say that one divides 72 by 6; obtaining 12, and in 12 years the capital will be doubled.”
• The Rule of 72 also gives rise to the rule of 144, which is used in exactly the same way as the Rule of 72, except 144 instead of 72.  This will tell you when the value will quadruple.
• The Rule of 72 doesn’t just apply to money; it actually applies to anything that grows.  For instance, if the average population growth rate for the planet Earth is 2%, then it will take just 72/2 = 36 years for the population of the Earth to double from the current 6.8 billion to 13.6 billion, then in another 36 years it will have doubled again to 27.2 billion!
• The world population growth rate was at its highest in the last 50 years in the 1960s when it hovered just over 2%.  Since then, it has been on a steady decline with the current annual population growth rate at just over 1%, so taking 72/1 = 72 years to double at that rate.
• Given population growth models through human history, it is estimated that there have existed around 100-115 billion humans in Earth’s history.  The idea that the total number of people alive today is more than the total number that have been alive in the past was based on the faulty premise put forth in the 1970s that 75% of all people that have ever lived were alive in the 1970s.  This has since been proven to be incorrect.
• Currently, the two largest countries, in terms of population, are China and India at 1.346 billion people and 1.21 billion people respectively, comprising around 37% of the entire global population.  China’s population growth rate is currently lower than the world-wide average; they are sitting at around .5%.  India’s population growth rate is currently above the world-wide average at just below 1.5%.
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